S. Ullman, The Interpretation Of Visual Motion (MIT Press, Cambridge, Mass., 1979).

A. P. Pentland, “Estimating illuminant direction to obtain shape from shading,” in Digest of Topical Meeting on Advances in Vision (Optical Society of America, Washington, D.C., 1980).

The change from a bandpass sensitivity for static images to a sensitivity that is a linear function of the rate of change in dynamic images may come about through summing of several bandpass channels, or it may be a change in the channels’ characteristics; the specific manner in which this change comes about is irrelevant to the task at hand. How this change in sensitivity happens is a question of the algorithm the visual system is employing and not a question of what the system is doing. What is of interest to us now is that under normal viewing conditions the human sensitivity to changes in image intensity is approximately a linear function of the rate of change and thus can be modeled by the first derivative of image intensity.

T. Cornsweet, Visual Perception (Academic, New York, 1970).

E. L. Kirnov, Spectral Properties of Natural Fountains, G. Belkov, translator, NRC of Canada Technical Translation 439 (National Research Council, Ottawa, Canada, 1971).

Here I refers to the image intensity at a point in the image, so that dI refers to the first derivative of image intensity. The context should make clear where, and in what direction, this derivative is taken.

That is, calculate dI by using a particular image orientation at each point in each of the images, and then calculate the mean of those dI for each image. Thus one number is calculated for each image and image orientation.

K. Stevens, “Surface perception from local analysis of texture and contour,” Ph.D. Thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1979).

This R is different from either the reflectance map1 or the bidirectional reflectance function, which describes how light is reflected for a particular L and V. The reflectance function defined here is a function of L, V, and N that, for particular values of L and V, generates the bidirectional reflectance function.

A. Pentland, “The visual inference of shape: computation from local features,” Ph.D. Thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1982).

Image irradiance is what is actually being discussed here, not image intensity, which is the measured image irradiance. I will continue to refer to image intensity, however, as that is customary in the psychological literature. The distinction is not important here, as the two can simply be assumed to be numerically equal.

This may be proved by noting that the surface normals on such an object are perpendicular to V at the boundary of such an object, and thus (given that the object is strictly convex) we may form a 1–1 onto map between the surface normals of the object and the Gaussian sphere, which satisfies Eq. (3).

In that regions of the image are the fundamental unit of estimation, this theory bears a resemblance to gestalt theories. This resemblance is heightened by the similarity of the gestalt concept of pragnanz, as used to provide constraint on the interpretation of an image, to the assumption of zero mean change in surface orientation, which was used to provide the constraint necessary to estimate the illuminant direction.